I think you should bet bigger on the flop to start with. $2 seems good, it's only 1bb now but think how much bigger the pot gets on the river actually.

As played, I think it's closer to a call than a fold. You will be beat sometimes but I expect QQ to 3bet preflop. You also don't have to be ahead of his range, I think Qx, flushdraws and straight draws make up enough of his range to call this shove.